In this work, we focus on the generalized coloring numbers introduced by Kierstead and Yang in 2003 [1]} which also characterize classes of bounded expansion [2]}. The generalized coloring numbers (consisting of the \(r\) -coloring numbers and the weak \(r\) -coloring numbers) extend the notion of degeneracy to neighborhoods of radius \(r\) .
The generalized coloring numbers are also deeply connected to other important graph properties.
In fact, the \(n\) -coloring number and the weak \(n\) -coloring number are equivalent to the treewidth and treedepth of a graph respectively.
Prior to the introduction of bounded expansion, the generalized coloring numbers were studied primarily in the context of the game chromatic number [1]}.
